After a bunch of testing I figured out what was causing the problem. The websites where I encountered the error were those that were set up on https servers (with SSL).
In an old article about reCAPTCHA 1 I learned that SSL sites need to have https links for these scripts. So I changed this one line of code in one of my SSL sites to this:
The only difference between these two lines is that the src for the second one begins with "https:" When I made this one simple change, the reCAPTCHA routine worked perfectly in all of the sites where the domains are on https servers.
A big thank you to Mudassar Ahmed Khan for the original code. His contributions in the ASP Snippets website are the best!